Subgroups — #26115076

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688855	C₂	1	4	yes	maximal
4	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15689029	C₄	1	2	yes	—
10	#15689021	—	2	2	no	—
11	#15688993	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₄₇	—	2	2	no	—
13	#15688855	—	2	8	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094024	—	2	4	no	—
20	C₄×Dic₉₄	—	2	4	no	—
21	C₄×Dic₉₄	—	2	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₄₇⋊C₄	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₄₇	—	2	2	no	—
37	C₄×C₈×C₄₇	D₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂²×Dic₄₇	—	2	2	no	—
45	C₂×Dic₉₄	—	2	4	no	—
46	C₂×Dic₉₄	—	2	4	no	—
47	C₂×Dic₉₄	—	2	2	no	—
48	C₂×Dic₉₄	M₁₆	1	4	yes	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	4	4	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	—	4	2	no	—
54	C₄²×C₄₇	SD₁₆	1	4	yes	—
55	C₄²×C₄₇	Q₁₆	1	4	yes	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₄⋊Dic₄₇	—	2	8	no	—
59	C₄⋊Dic₄₇	—	2	4	no	—
60	C₄⋊Dic₄₇	—	4	4	no	—
61	Dic₄₇⋊C₄	—	4	4	no	—
62	C₄×Dic₄₇	—	4	2	no	—
63	C₄×Dic₄₇	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	2	no	—
65	C₂×C₄₇⋊C₈	—	4	4	no	—
66	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	#103	1	4	yes	—
73	C₂×C₄×C₄₇	#103	1	4	yes	—
74	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₄₇	#121	1	2	yes	—
76	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₄₇	#131	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
82	C₂×Dic₄₇	—	2	4	no	—
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	4	2	no	—
85	C₂×Dic₄₇	—	4	2	no	—
86	C₂×Dic₄₇	—	4	4	no	—
87	Dic₉₄	—	2	8	no	—
88	Dic₉₄	—	4	4	no	—
89	C₈×C₄₇	—	4	4	no	—
90	C₄₇⋊C₈	—	8	2	no	—
91	#9643	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	#390	1	1	yes	characteristic
93	C₂²×C₄₇	#407	1	1	yes	characteristic
94	C₂²×C₄₇	#408	1	1	yes	characteristic
95	C₂²×C₄₇	C₂×#103	1	2	yes	—
96	C₂²×C₄₇	#421	1	2	yes	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	2	4	no	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	#435	1	4	yes	—
101	C₄×C₄₇	#487	1	2	yes	—
102	C₄×C₄₇	#492	1	1	yes	characteristic
103	C₄×C₄₇	#494	1	1	yes	characteristic
104	Dic₄₇	—	4	4	no	—
105	Dic₄₇	—	8	2	no	—
106	C₄×C₄⋊Q₈	—	47	1	no	—
107	#1842	—	47	1	no	—
108	#1744	D₄₇	1	1	yes	characteristic
109	C₂×Q₈⋊C₈	—	47	4	no	—
110	C₂×C₄₇	#1481	1	2	yes	—
111	C₂×C₄₇	#1487	1	2	yes	—
112	C₂×C₄₇	#1955	1	1	yes	characteristic
113	C₂×C₄₇	#1956	1	1	yes	characteristic
114	C₂×C₄₇	#1963	1	1	yes	characteristic
115	C₂×C₄⋊Q₈	—	47	1	no	—
116	C₂×C₄×Q₈	—	47	2	no	—
117	C₂×C₄⋊C₈	—	47	2	no	—
118	C₂×C₄⋊C₈	—	94	2	no	—
119	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
120	#391	—	47	2	no	—
121	#389	—	94	2	no	—
122	#384	—	47	1	no	—
123	C₄×C₄⋊C₄	—	94	2	no	—
124	C₄³	C₂×D₄₇	1	1	yes	characteristic
125	#336	—	2	2	no	—
126	Q₈⋊C₈	—	94	8	no	—
127	C₄₇	#9643	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	47	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	94	2	no	—
131	C₄⋊Q₈	—	94	4	no	—
132	C₄×Q₈	—	94	4	no	—
133	C₄×Q₈	—	94	4	no	—
134	C₂×C₄⋊C₄	—	47	2	no	—
135	C₂×C₄⋊C₄	—	94	2	no	—
136	C₂×C₄⋊C₄	—	94	2	no	—
137	C₂×C₄⋊C₄	—	94	2	no	—
138	C₂×C₄²	—	94	2	no	—
139	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
141	C₂×C₄²	D₁₈₈	1	2	yes	—
142	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
143	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
144	C₄⋊C₈	—	94	4	no	—
145	C₄⋊C₈	—	188	4	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	D₁₈₈	1	4	yes	—
148	#95	—	94	4	no	—
149	C₂×Q₈	—	47	4	no	—
150	C₂×Q₈	—	94	2	no	—
151	C₂×Q₈	—	94	4	no	—
152	C₂×Q₈	—	94	4	no	—
153	C₂²×C₄	—	94	2	no	—
154	C₂²×C₄	—	94	2	no	—
155	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
156	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	188	2	no	—
163	C₂×C₈	—	188	4	no	—
164	C₄⋊C₄	—	94	8	no	—
165	C₄⋊C₄	—	94	4	no	—
166	C₄⋊C₄	—	188	4	no	—
167	C₄⋊C₄	—	188	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	188	2	no	—
172	C₄²	—	188	2	no	—
173	C₄²	#10625325	1	4	yes	—
174	C₄²	Dic₁₈₈	1	4	yes	—
175	C₄²	C₂×D₁₈₈	1	2	yes	—
176	C₄²	#10625350	1	2	yes	—
177	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	Q₈	—	94	8	no	—
179	Q₈	—	188	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	94	4	no	—
186	C₂×C₄	—	188	2	no	—
187	C₂×C₄	—	188	4	no	—
188	C₂×C₄	—	188	2	no	—
189	C₂×C₄	—	188	2	no	—
190	C₂×C₄	#13093949	1	4	yes	—
191	C₂×C₄	#13093949	1	4	yes	—
192	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
193	C₂×C₄	#13093998	1	2	yes	—
194	C₂×C₄	#13094012	1	1	yes	characteristic
195	C₂×C₄	#13094021	1	1	yes	characteristic
196	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
197	C₂×C₄	#13094032	1	2	yes	—
198	C₂×C₄	C₂×#10625325	1	2	yes	—
199	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	376	2	no	—
202	C₂²	#15689025	1	1	yes	characteristic
203	C₂²	#15689086	1	1	yes	characteristic
204	C₂²	#15689091	1	1	yes	characteristic
205	C₂²	C₂×#13093949	1	2	yes	—
206	C₂²	#15689141	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	376	2	no	—
211	C₄	—	188	4	no	—
212	C₄	#15689196	1	4	yes	—
213	C₄	#15689474	1	2	yes	—
214	C₄	#15689496	1	1	yes	characteristic
215	C₄	#15689503	1	1	yes	characteristic
216	C₂	#20241792	1	2	yes	—
217	C₂	#20241818	1	2	yes	—
218	C₂	#20244186	1	1	yes	characteristic
219	C₂	#20244190	1	1	yes	characteristic
220	C₂	#20244220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#9643
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
127. C₄₇	#9643	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#13094024	2
#20243540	2
C₂×#15688855	2
C₄₇×#1744	2
#9643	47